Lattice Points Research Articles

Sequential good lattice point sets for computer experiments

Sequential Latin hypercube designs have recently received great attention for computer experiments. Much of the work has been restricted to invariant spaces. The related systematic construction methods are inflexible while algorithmic methods are ineffective for large designs. For such designs in space contraction, systematic construction methods have not been investigated yet. This paper proposes a new method for constructing sequential Latin hypercube designs via good lattice point sets in a variety of experimental spaces. These designs are called sequential good lattice point sets. Moreover, we provide fast and efficient approaches for identifying the (nearly) optimal sequential good lattice point sets under a given criterion. Combining with the linear level permutation technique, we obtain a class of asymptotically optimal sequential Latin hypercube designs in invariant spaces where the $L_1$-distance in each stage is either optimal or asymptotically optimal. Numerical results demonstrate that the sequential good lattice point set has a better space-filling property than the existing sequential Latin hypercube designs in the invariant space. It is also shown that the sequential good lattice point sets have less computational complexity and more adaptability.

Fermionic and Bosonic Partition Functions at Imaginary Chemical Potential as Bloch Functions

In this work it is pointed out that the phase transitions of the <i>d</i>+1 Gross-Neveu (fermionic) and <I>CP<SUP>N</SUP></I><sup>−1</sup> (bosonic) models at finite temperature and imaginary chemical potential can be mapped to transformations of Hubbard-like regular hexagonal to square lattice with the intermediate steps to be specific surfaces (irregular hexagonal kind) with an ordered construction based on the even indexed Bloch-Wigner-Ramakrishnan polylogarithm function. The zeros and extrema of the Clausen <i>Cl<sub>d</sub></i>(<i>θ</i>) function play an important role to the analysis since they allow us not only to study the fermionic and bosonic theories and their phase transitions but also the possibility to explore the existence of conductors arising from the correspondence between the partition functions of the two models and the Bloch and Wannier functions that play a crucial role in the tight-binding approximation in solid state physics. The main aim of this work is not only to unveil the relevance of the canonical partition functions of a fermionic and a bosonic model to Bloch states by using an imaginary chemical potential but also to examine the overlap between two Bloch wave-functions that differ by a lattice momentum that calculates the momentum transfer of a Bloch wave during the interaction with a lattice point of a hexagonal construction.

Lattice points on a curve via ℓ2 decoupling

This paper extends Bombieri and Pila’s estimate of lattice points on curves to arbitrary finite sets by incorporating considerations of minimal separation and the doubling constant. We derive the estimate by establishing the [Formula: see text] decoupling inequality for non-degenerate curves in [Formula: see text]. Additionally, we review the curve-lifting method introduced in Bombieri and Pila’s work and establish the estimate of lattice points in the neighborhood of a planar curve.

Hyperbolic lattice point counting in unbounded rank

Hyperbolic lattice point counting in unbounded rank

A new invariant of lattice polytopes

The maximal degree of monomials belonging to the unique minimal system of monomial generators of the canonical module $\omega (K[\mathcal{P}])$ of the toric ring $K[\mathcal{P}]$ defined by a lattice polytope $\mathcal{P}$ will be studied. It is shown that if $\mathcal{P}$ possesses an interior lattice point, then the maximal degree is at most $\dim \mathcal{P} - 1$, and that this bound is the best possible in general.

On the number of lattice points in thin sectors

On the circle of radius R centred at the origin, consider a “thin” sector about the fixed line y=αx\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y = \\alpha x$$\\end{document} with edges given by the lines y=(α±ϵ)x\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$y = (\\alpha \\pm \\epsilon ) x$$\\end{document}, where ϵ=ϵR→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon = \\epsilon _R \\rightarrow 0$$\\end{document} as R→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ R \\rightarrow \\infty $$\\end{document}. We establish an asymptotic count for Sα(ϵ,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\\alpha }(\\epsilon ,R)$$\\end{document}, the number of integer lattice points lying in such a sector. Our results depend both on the decay rate of ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document} and on the rationality/irrationality type of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}. In particular, we demonstrate that if α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is Diophantine, then Sα(ϵ,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\\alpha }(\\epsilon ,R)$$\\end{document} is asymptotic to the area of the sector, so long as ϵRt→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon R^{t} \\rightarrow \\infty $$\\end{document} for some t<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ t<2 $$\\end{document}.

Poissonian pair correlation for directions in multi-dimensional affine lattices and escape of mass estimates for embedded horospheres

Abstract We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author [The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems. Ann. of Math. (2)172 (2010), 1949–2033], and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname {SL}(d,\mathbb {R})$ -horospheres in the space of affine lattices.

Numerical Study on Permeability of Reconstructed Porous Concrete Based on Lattice Boltzmann Method

The reconstruction of the porous media model is crucial for researching the mesoscopic seepage characteristics of porous concrete. Based on a self-compiled MATLAB program, a porous concrete model was modeled by controlling four parameters (distribution probability, growth probability, probability density, and porosity) with clear physical meanings using a quartet structure generation set (QSGS) along with the lattice Boltzmann method (LBM) to investigate permeability. The rationality of the numerical model was verified through Poiseuille flow theory. The results showed that the QSGS model exhibited varied pore shapes and disordered distributions, resembling real porous concrete. Seepage velocity distribution showed higher values in larger pores, with flow rates reaching up to 0.012 lattice point velocity. The permeability–porosity relationship demonstrated high linearity (the Pearson correlation coefficient is 0.92), consistent with real porous concrete behavior. The integration of QSGS-LBM represents a novel approach, and the research results can provide new ideas and new means for subsequent research on the permeability of porous concrete or similar porous medium materials.

Two-dimensional silicon-based dielectric column photonic crystal point defect microcavity Neural network modelling

In order to solve the problem that it is difficult to predict the microcavity photonic energy bands of point defects in two-dimensional silicon-based dielectric column photonic crystals, this paper proposes a method to predict the microcavity photonic energy bands of point defects in two-dimensional silicon-based photonic crystals by using an artificial neural network model. In this paper, the energy band structure of triangular lattice point defects at different radii is calculated using MPB, and a neural network model is established based on this data and the accuracy of the established model is verified.

Manipulation of Weyl Points in Reciprocal and Nonreciprocal Mechanical Lattices.

We introduce feedback-measurement technologies to achieve flexible control of Weyl points and conduct the first experimental demonstration of Weyl type I-II transition in mechanical systems. We demonstrate that non-Hermiticity can expand the Fermi arc surface states from connecting Weyl points to Weyl rings, and lead to a localization transition of edge states influenced by the interplay between band topology and the non-Hermitian skin effect. Our findings offer valuable insights into the design and manipulation of Weyl points in mechanical systems, providing a promising avenue for manipulating topological modes in non-Hermitian systems.

RDDA method for characterization of photonic nanojets.

This paper presents a reduced discrete dipole approximation (RDDA) method as a flexible and powerful tool for computing the field distribution of photonic nanojets (PNJs) based on superposition of the induced dipole radiation fields with a low number of lattice points at the surface of microparticles. Microparticles with spherical and ellipsoidal shapes have been studied using the RDDA method to produce PNJs. The incident beam polarization and filling factor effects in the PNJs' shape and confinement strength have been characterized by quality factor Q. The best confined PNJs have been found in microellipsoids with a specific range of the shape parameter and filling factor values.

A characterisation of linear repetitivity for cut and project sets with general polytopal windows

The cut and project method is a central construction in the theory of Aperiodic Order for generating quasicrystals with pure point diffraction. Linear repetitivity (LR) is a form of ideal regularity of aperiodic patterns. Recently, Koivusalo and the present author characterised LR for cut and project sets with convex polytopal windows whose supporting hyperplanes are commensurate with the lattice, the weak homogeneity property. For such cut and project sets, we show that LR is equivalent to two properties. One is a low complexity condition, which may be determined from the cut and project data by calculating the ranks of the intersections of the projection of the lattice to the internal space with the subspaces parallel to the supporting hyperplanes of the window. The second condition is that the projection of the lattice to the internal space is Diophantine (or ‘badly approximable’), which loosely speaking means that the lattice points in the total space stay far from the physical space, relative to their norm. We review then extend these results to non-convex and disconnected polytopal windows, as well as windows with polytopal partitions producing cut and project sets of labelled points. Moreover, we obtain a complete characterisation of LR in the fully general case, where weak homogeneity is not assumed. Here, the Diophantine property must be replaced with an inhomogeneous analogue. We show that cut and project schemes with internal space isomorphic to Rn⊕G⊕Zr, for G finite Abelian, can, up to MLD equivalence, be reduced to ones with internal space Rn, so our results also cover cut and project sets of this form, such as the (generalised) Penrose tilings.

Automated target placement for VMAT lattice radiation therapy: enhancing efficiency and consistency

Objective. An algorithm was developed for automated positioning of lattice points within volumetric modulated arc lattice radiation therapy (VMAT LRT) planning. These points are strategically placed within the gross tumor volume (GTV) to receive high doses, adhering to specific separation rules from adjacent organs at risk (OARs). The study goals included enhancing planning safety, consistency, and efficiency while emulating human performance. Approach. A Monte Carlo-based algorithm was designed to optimize the number and arrangement of lattice points within the GTV while considering placement constraints and objectives. These constraints encompassed minimum spacing between points, distance from OARs, and longitudinal separation along the z-axis. Additionally, the algorithm included an objective to permit, at the user’s discretion, solutions with more centrally placed lattice points within the GTV. To validate its effectiveness, the automated approach was compared with manually planned treatments for 24 previous patients. Prior to clinical implementation, a failure mode and effects analysis (FMEA) was conducted to identify potential shortcomings. Main results. The automated program successfully met all placement constraints with an average execution time (over 24 plans) of 0.29 ±0.07 min per lattice point. The average lattice point density (# points per 100 c.c. of GTV) was similar for automated (0.725) compared to manual placement (0.704). The dosimetric differences between the automated and manual plans were minimal, with statistically significant differences in certain metrics like minimum dose (1.9% versus 1.4%), D5% (52.8% versus 49.4%), D95% (7.1% versus 6.2%), and Body-GTV V30% (20.7 c.c. versus 19.7 c.c.). Significance. This study underscores the feasibility of employing a straightforward Monte Carlo-based algorithm to automate the creation of spherical target structures for VMAT LRT planning. The automated method yields similar dose metrics, enhances inter-planner consistency for larger targets, and requires fewer resources and less time compared to manual placement. This approach holds promise for standardizing treatment planning in prospective patient trials and facilitating its adoption across centers seeking to implement VMAT LRT techniques.

Visible lattice points along curves in higher dimensional random walks

Let k≥2 be an integer, and α=(α1,⋯,αk) with 0<α1,⋯,αk<1 and α1+⋯+αk=1, this paper concerns the generalized visibility of lattice points visited by an α-random walk in Zk. Specifically, for b=(b1,⋯,bk)∈Nk with gcd⁡(b1,⋯,bk)=1, we focus on the lattice points in the walk which are b-visible from a set of N watch-points, and we show that, almost surely, the proportion of b-visible steps is ∏p(1−N/ps(b)), where p runs over all primes and s(b):=b1+⋯+bk. This paper generalizes our previous work for the visible lattice points in higher dimensional random walks.

Lattice points in slices of prisms

Abstract We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov; moreover, they coincide with polymatroids satisfying the strong exchange property up to an affinity. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, via an extension of a result by Early and Kim, we give a combinatorial interpretation for all the coefficients of the $h^*$ -polynomial. All of our results provide a combinatorial understanding of the Hilbert functions and the h-vectors of all algebras of Veronese type, a problem that had remained elusive up to this point. A variety of applications are discussed, including expressions for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; some extensions of Laplace’s result on the combinatorial interpretation of the volume of the hypersimplex; a multivariate generalization of the flag Eulerian numbers and refinements; and a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.

Pair correlations of logarithms of complex lattice points

We study the correlations of pairs of complex logarithms of Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}$$\\end{document}-lattice points in C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}$$\\end{document} at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighbourhood to itself in the Bianchi orbifold PSL2(Z[i])\\HR3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {PSL}}_2({\\mathbb {Z}}[i]) \\backslash {{{\\mathbb {H}}}^3_{\\mathbb {R}}}$$\\end{document}.

Asymptotic distribution of nodal intersections for ARW against a surface

We investigate Gaussian Laplacian eigenfunctions (Arithmetic Random Waves) on the three-dimensional standard flat torus, in particular the asymptotic distribution of the nodal intersection length against a fixed regular reference surface. Expectation and variance have been addressed by Maffucci [Ann. Henri Poincaré 20(11), 3651–3691 (2019)] who found that the expected length is proportional to the square root of the eigenvalue times the area of the surface, while the asymptotic variance only depends on the geometry of the surface, the projected lattice points being equidistributed on the two-dimensional unit sphere in the high-energy limit. He also noticed that there are “special” surfaces, so-called static, for which the variance is of smaller order; however he did not prescribe the precise asymptotic law in this case. In this paper, we study second order fluctuations of the nodal intersection length. Our first main result is a Central Limit Theorem for “generic” surfaces, while for static ones, a sphere or a hemisphere e.g., our main results are a non-Central Limit Theorem and a precise asymptotic law for the variance of the nodal intersection length, conditioned on the existence of so-called well-separated sequences of Laplacian eigenvalues. It turns out that, in this regime, the nodal area investigated by Cammarota [Trans. Am. Math. Soc. 372(5), 3539–3564 (2019)] is asymptotically fully correlated with the length of the nodal intersections against any sphere. The main ingredients for our proofs are the Kac-Rice formula for moments, the chaotic decomposition for square integrable functionals of Gaussian fields, and some arithmetic estimates that may be of independent interest.

Branching Random Walks in a Random Killing Environment with a Single Reproduction Source

We consider a continuous-time branching random walk on Z in a random non-homogeneous environment. The process starts with a single particle at initial time t=0. This particle can walk on the lattice points or disappear with a random intensity until it reaches the certain point, which we call the reproduction source. At the source, the particle can split into two offspring or jump out of the source. The offspring of the initial particle evolves according to the same law, independently of each other and the entire prehistory. The aim of this paper is to study the conditions for the presence of exponential growth of the average number of particles at every lattice point. For this purpose, we investigate the spectrum of the random evolution operator of the average particle numbers. We derive the condition under which there is exponential growth with probability one. We also study the process under the violation of this condition and present the lower and upper estimates for the probability of exponential growth.

Continuous-wave operation of 1550 nm low-threshold triple-lattice photonic-crystal surface-emitting lasers

Benefitting from narrow beam divergence, photonic crystal surface-emitting lasers are expected to play an essential role in the ever-growing fields of optical communication and light detection and ranging. Lasers operating with 1.55 μm wavelengths have attracted particular attention due to their minimum fiber loss and high eye-safe threshold. However, high interband absorption significantly decreases their performance at this 1.55 μm wavelength. Therefore, stronger optical feedback is needed to reduce their threshold and thus improve the output power. Toward this goal, photonic-crystal resonators with deep holes and high dielectric contrast are often used. Nevertheless, the relevant techniques for high-contrast photonic crystals inevitably complicate fabrication and reduce the final yield. In this paper, we demonstrate the first continuous-wave operation of 1.55 μm photonic-crystal surface-emitting lasers by using a ‘triple-lattice photonic-crystal resonator’, which superimposes three lattice point groups to increase the strength of in-plane optical feedback. Using this geometry, the in-plane 180° coupling can be enhanced threefold compared to the normal single-lattice structure. Detailed theoretical and experimental investigations demonstrate the much lower threshold current density of this structure compared to ‘single-lattice’ and ‘double-lattice’ photonic-crystal resonators, verifying our design principles. Our findings provide a new strategy for photonic crystal laser miniaturization, which is crucial for realizing their use in future high-speed applications.

Strain-controlled charge and spin current rectifications in spin–orbit coupled graphene nano-ribbon: A new proposition

In this work, we investigate the possibilities of performing charge and spin current rectifications using graphene nano-ribbon in the presence of Rashba spin–orbit (SO) interaction. More specifically, we explore the specific role of mechanical strain on these two different types of current rectifications. The system is simulated by a tight-binding framework, where all the results are worked out based on the standard Green’s function formalism. In order to have current rectification, an asymmetry is required, which is incorporated through uncorrelated disorder among the constituent lattice points. From our extensive numerical analysis, we find that reasonably large charge and spin current rectifications can be obtained under strained conditions, and all the physical pictures are valid for a broad range of tight-binding parameters. The rectification properties are studied mostly for zigzag graphene nano-ribbons; however, an armchair ribbon is also taken into account for a clear comparison. Our work may provide a new direction of getting strain-controlled current rectifications in similar kinds of other physical systems as well.